A line is simply connected but a circle is not. Does this mean that the one-point compactification of the real line is not simply connected?

Question

I was reading about end compactification, following up my interest in compactification of the real line, Homeomorphism of a compact real line to the real line.

Question 1&2: Is a Warsaw circle path connected? Is it simply connected?

I was reading an article that said,the Warsaw circle is path connected, the topologist's sine circle is path-connected but it's not locally path-connected, and another article says it is not. (See http://mathforum.org/kb/message.jspa?messageID=4032108. See also: http://sci.tech-archive.net/Archive/sci.math/2005-10/msg02190.html)

So, now I am confused. The two end points of the topologist's sine curve are connected, so the Warsaw circle should be path connected. However, it should not be simply connected. Did I miss something here? Thanks.

Question 3: Is the one point compactification of the real line simply connected?

The above disagreement between articles got me thinking of a one-point compactification of a the real-line. The real line is locally connected and path connected and simply connected, correct? However, it's one point compactification is homeomorphic to a circle and a circle is not simply-connected in two dimensions. (That is, there is no way to contract the path to a point, though there is for the real line.)

Does this mean that the one-point compactification of the real line is not simply connected even though the real line is? (I was reading: https://en.wikipedia.org/wiki/Simply_connected_space)

Answer 1

There are two different spaces called the Warsaw circle: (1) https://en.wikipedia.org/wiki/Continuum_(topology)#/media/File:Warsaw_Circle.png and (2) http://ncatlab.org/nlab/files/warsaw.pdf.

Basically, in (1), the 'special' point - which I'll call "$*$" - is only badly behaved on one side: the circle is smooth immediately to the left of $*$, and chaotic immediately to the right. This version is indeed path connected but not simply connected.

In (2), though, $*$ is bad on both sides. (2) is thus not path connected, since there's no path connecting $*$ to any other point (in (1), such a path can be found by going around the left).

Answer 2

The one point compactification of the real line is the circle. (If you remove one point of a circle, you obtain an open interval which diffeomorphic to the real line).